This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the questions: QUESTION 1 1.1.1. What is the name of the graph given above? Step 1: Identify the form of the function. The function f(x) = -2x^2 + 8 is a quadratic function of the form ax^2 + bx + c. The graph of a quadratic function is a parabola. The name of the graph is a parabola. 1.1.2. Calculate both the x-intercepts and y-intercepts of the graph above. Step 1: Calculate the x-intercepts by setting f(x) = 0. -2x^2 + 8 = 0 8 = 2x^2 4 = x^2 x = ±sqrt(4) x = ± 2 The x-intercepts are (-2, 0) and (2, 0). Step 2: Calculate the y-intercept by setting x = 0. f(0) = -2(0)^2 + 8 f(0) = 8 The y-intercept is (0, 8). 1.1.3. What is the axis of symmetry of the graph. Step 1: For a quadratic function f(x) = ax^2 + bx + c, the axis of symmetry is given by the formula x = -(b)/(2a). For f(x) = -2x^2 + 8, we have a = -2 and b = 0. x = -(0)/(2(-2)) x = 0 The axis of symmetry is x = 0. 1.1.4. Draw the graph of f(x) and show the x-intercepts and the y-intercepts on it. To draw the graph: • Plot the x-intercepts at (-2, 0) and (2, 0). • Plot the y-intercept at (0, 8). This is also the vertex of the parabola since the axis of symmetry is x=0. • Since the coefficient of x^2 is negative (-2), the parabola opens downwards. • Draw a smooth curve connecting these points, opening downwards and symmetric about the y-axis (x=0). 1.2. The following diagram represents the graph of f(x) = (a)/(x) + q. A(4;2) is a point on the graph. The question asks to determine the values of a and q. Step 1: Identify the horizontal asymptote from the graph to find q. From the graph, the horizontal asymptote is the line that the graph approaches as x tends to positive or negative infinity. This line is y=1. For a function of the form f(x) = (a)/(x) + q, the horizontal asymptote is y=q. Therefore, q = 1 Step 2: Use the given point A(4, 2) and the value of q to find a. Substitute x=4, f(x)=2, and q=1 into the equation f(x) = (a)/(x) + q. 2 = (a)/(4) + 1 Step 3: Solve for a. 2 - 1 = (a)/(4) 1 = (a)/(4) a = 1 × 4 a = 4 The values are a = 4 and q = 1. That's 2 down. 3 left today — send the next one.